Spatial dynamics of water and nitrogen management in irrigated agriculture.

Spatial variability is a dynamic extension of the static model proposed by Seginer (1978) and used subsequently in Feinerman, Letey, and Vaux (1983), Dinar, Letey, and Knapp (1985), and Berck and Helfand (1990). The key concept is a water infiltration coefficient giving the fraction of field-average water depth infiltrating at a point in the field. At a particular point in the field, the amount of water infiltrating into the root zone at time t is [w.sub.t]([beta]) = [beta][[bar.w].sub.t] where [beta] [member of] [0, [infinity]] is the water infiltration coefficient. [beta] is distributed over the field according to a spatial density function, f([beta]), with mean E[[beta]] = 1, and standard deviation SD[[beta]] that depends on the type of irrigation system.

Field-level relationships for yield and nitrogen emissions are:

(2) [[bar.t].sub.t] = [[integral].sup.[infinity].sub.0] [y.sub.t]([beta])f([beta])d[beta] [[bar.n].sub.et] = [[integral].sup.[infinity].sub.0] [n.sub.et]([beta])f([beta])d[beta]

where [y.sub.t]([beta]) and [n.sub.et]([beta]) are plant-level yield [Mg/ha] and nitrogen emissions [kg/ha], respectively. Thus field-level crop yield and nitrogen emissions are plant-level yield and emissions integrated over the field according to the spatial density function for water infiltration. Plant-level production functions for yield and nitrogen emissions are [y.sub.t]([beta]) = [g.sub.y][n.sub.t]([beta]), [w.sub.t]([beta]), [n.sub.at]([beta])] and [n.sub.et]([beta]) = [g.sub.e][[n.sub.t][beta]), [w.sub.t]([beta]), [n.sub.at]([beta])], respectively, where [n.sub.t] is inorganic soil nitrogen [kg/ha] at the beginning of period t, and [n.sub.at] is applied nitrogen. At the plant-level, crop yield and nitrogen emissions, specified as leaching below the rootzone, depend on initial soil nitrogen, water infiltration, and nitrogen applications at points in the field characterized by [beta].

Soil nitrogen dynamics or carryover dynamics (Segarra et al. 1989) for a given [beta] are

(3) [n.sub.t+1]([beta]) = [g.sub.n][[n.sub.t]([beta]), [w.sub.t]([beta]), [n.sub.at]([beta])]

indicating dependence on the same variables as plant-level crop yield and nitrogen emissions. Initial soil nitrogen in period 1 is assumed constant across the field [[n.sub.1]([beta]) = [[bar.n].sub.1]], and nitrogen is applied uniformly across the field [[n.sub.at]([beta]) = [[bar.n].sub.at]], the latter assumption following from the use of mechanical/chemical fertilizer applications consistent with irrigated agriculture. The model can be modified to make plant-level fertilizer applications proportional to infiltrated irrigation water; however, this is not pursued here. For computational tractability in the dynamic optimization model, the spatial density support is discretized into a series of intervals, each with a specified [beta] value and representing a fraction of the field as computed from the spatial density function. A useful interpretation is that the field is divided into a finite number of plots each with a specified [beta] value and area. (6)

Control variables are field-level applied water [[bar.w].sub.t] and nitrogen [[bar.n].sub.at], and state variables are nitrogen carryover for each of the discrete grid intervals for the [beta] infiltration coefficients. The dynamic optimization problem is solved using the GAMS CONOPT nonlinear optimization procedure. To eliminate endpoint effects, the optimization routine is implemented as a running horizon problem in which a sequence of finite-horizon optimization problems are solved with a thirty-year time horizon, each starting from the states resulting from the first period of the previous solution and retaining only the first period results from each for the final solution.

Economic Data and Crop-Water-Nitrogen Production Function

The empirical application is corn production in Yolo County, California with a traditional (furrow one-half mile) irrigation system. Maximum corn yield is 12.02 Mg/ha, with a price of $102.02 [[Mg.sup.-1]]. Production costs include costs such as seed, land preparation, and machinery but do not include those associated with water, nitrogen fertilizer, land and management, and cash overhead (UCCE 2004). Irrigation system data are from University of California Committee of Consultants (UCCC 1988). Combined, amortized nonwater production costs are $673 [ha.sup.-1], baseline nitrogen fertilizer costs are $0.59 [kg.sup.-1], and baseline water costs are $0.64 [[ha cm].sup.-1]. We assume a discount rate of 5% with all economic data inflation-adjusted to 2003 dollars.

The infiltration coefficients [beta] are distributed lognormally over the field with E[[beta]] = 1 for mass balance. The baseline results assume a Christensen Uniformity Coefficient (CUC) of 0.77, where CUC is a widely used measure of nonuniformity in the irrigation engineering literature, and calculated as 1 - [[integral].sup.[infinity].sub.0] [absolute value of [beta] - 1] f ([beta])d[beta]. SD[[beta]] was estimated so that the CUC = 0.77 under the lognormal [beta] distribution. This distribution is discretized into 11 possible [beta] values, each with an associated fraction of the field computed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where f is the lognormal density and [[DELTA].sub.i], i = 1,11, is a partition of [0, [infinity]] containing the discrete [beta] values. This model can be interpreted as 11 subareas of the field, each characterized by a [beta] value, constituting a specified fraction of the field, and with an associated soil nitrogen state variable.

A classic work on water-nitrogen production functions is Hexum and Heady (1978). Although they investigate several functional forms, they settle on polynomials (including fractional powers) as a useful functional form. Ackello-Ogutu, Paris, and Williams (1985), among others, point out that polynomials generally do not fit qualitative agronomic theory and evidence: they have a point maximum instead of a plateau maximum, allow more substitution than is warranted by the data, and imply excessive input usage. Moreover, von-Liebig functions demonstrate superior data fit relative to polynomials and other traditional smooth production functions (Ackello-Ogutu, Paris, and Williams 1985; Grimm, Paris, and Williams 1987; Paris 1992). However, a recent sophisticated statistical analysis by Berck, Geoghegan, and Stohs (2000) rejects both the von-Liebig formulation as well as the non-substitution hypothesis. Taken together, these results leave open the appropriate form for plant-level production functions.

[FIGURE 1 OMITTED]

Additional concerns arise at the field-level with spatial variability. As outlined in Lanzer and Paris (1981; figure 1), a general conceptual model of yield production functions exhibits convex-concave behavior initially, followed by a yield plateau and then possibly a yield decline. In the uniform case, only the concave portion is economically relevant, hence functional forms constituting local approximations (e.g., Taylor series approximations via polynomials) may be reasonable as the optimization model can appropriately bound the inputs. In the spatial case, though, some parts of the field likely receive input levels in the convex (increasing returns to scale) portion, while other parts receive excess input levels leading to yield declines. Consequently, functions with desirable global properties and data fit are necessary, raising additional issues to those debated in the literature. Polynomials with any reasonable order and von-Liebig functions are unlikely to perform well globally even if they are reasonable locally.

To overcome some of these difficulties, we develop a production function system specified by several component functions representing the major flows and processes in the plant-water-soil system. One reason for the system approach rather than the approach used in much of the literature (e.g., Johnson, Adams, and Perry 1991; Vickner et al. 1998) is that a system approach can capture yield-depressing effects associated with excess water infiltration in a logical fashion while still allowing individual component functions to be estimated with classical properties. We also utilize functional forms that exhibit convex-concave behavior and plateau maximums. These functional forms effectively place upper and lower bounds on the levels for individual variables. In combination with multiplicative functions such as Mitscherlich-Baule (Paris 1992), this system allows for input substitution consistent with Berck, Geoghegan, and Stohs (2000), yet subject to limits consistent with the classic findings of Paris and others.

The plant-level production system was estimated for corn using an unusually rich data set from Tanji et al. (1979) (see also Pang, Letey, and Wu 1997a, b). The experimental data consist of two years of corn field trials at a University of California-Davis site from October 1974 through September 1976. The trials measure the effects of nitrogen and water applications rates on yields, nitrogen uptake, inorganic soil nitrogen levels, nitrate emissions, and organic nitrogen mineralization. The experiment provides data beyond that typically used in the agricultural production economics literature (e.g., Hexum and Heady 1978 as used in Berck, Geoghegan, and Stohs 2000), and is key to the analysis as it allows estimation and testing of the system model without resorting to hidden variables and speculative functional forms. It should be emphasized that while these field experiments were performed in the mid 1970s, recent articles in the soil science literature still calibrate to this data (Pang, Letey, and Wu 1997a,b).